In this section, we compare the positioning performances of two different vacuum sources using PID and LQR control methods. First, we perform experiments with no external load, and then we add a constant load of 500 g to the system.
3.1. Position Performances of the Bellow PMA by Utilizing Indirect Vacuum Control
The PMA responses in no-load conditions are recorded for the PID and LQR approaches for different reference values (5, 10, 20, 30, 40 and 50 mm). Please keep in mind that, while it has been demonstrated that the working range of the employed PMA can be extended to approximately 60 mm [
10], we limit the working range to 50 mm in this paper to avoid the contact of the inner supporting rings which can introduce additional nonlinearity to the setup.
As previously stated, in the case of indirect vacuum control (type A—please refer to
Section 2.1 and
Figure 1a), the proportional regulator varies the output pressure at the input rail of the used ejector, creating vacuum at its output. In this case, we experimentally confirmed that there is some dead-band present in the PMA response, as it can be observed from
Figure 7. It can also be seen that the PMA has very similar dynamical behavior for both PID and LQR, though PID shows slightly faster dynamics for 5, 20 and 40 mm reference values. Steady-state error is, however, always considerably lower when LQR is used and this result is mainly limited by the used feedback sensor. In some cases, LQR outperforms PID by an order of magnitude, while the error is kept to a few tens of micrometers for all reference values. However, for both control typologies, there is a tendency for steady-state error to increase with higher reference values (except for the highest reference in case of LQR). When compared to similar research in this field [
19], where error is measured in millimeters, this can be considered as a very good result, especially given that it was achieved on a highly nonlinear pneumatic system. Please note that the achieved positioning results are also limited by the properties of the used sensor (see also
Section 2). Both controllers provide responses without overshoot, but the jitter effect is pronounced when PID is used. Similar to previous experiments, the muscle does not return exactly to the reference position, since all experiments are performed without the application of an external load.
Table 2 summarizes the values of rising times and steady-state errors.
Given the fact that there is always a certain amount of dead-band present in the responses when indirect vacuum control is used, we analyze the behavior of the system by employing direct vacuum control in the following section.
3.2. Position Performances of the Bellow PMA by Utilizing Direct Vacuum Control
In this section, we conduct the experiments by using the type B pneumatic system (see
Figure 1c), which allows for direct vacuum control at the output of the ejector. This configuration also allows for much faster valve switching (approximately 3 Hz frequency). Please keep in mind that one of the disadvantages of this system is that it consumes more energy due to the need for constant vacuum supply to the valve.
Since the system’s hardware has been considerably modified, the optimal gains of both utilized controllers had to be adjusted. The PID parameters that were adopted are as follows:
KP = 0.295,
KI = 0.035 and
KD = 3. The vector with LQR gains, on the other hand, is calculated to be K = [0.2 0.002]. As with the type A system, we also calculate an additional pregain term in this case to allow for the elimination of the steady-state error. As shown in
Table 3, the pregain term (
Figure 8) is defined as a third order polynomial with coefficients that again depend on the motion direction.
Figure 9 depicts and compares the experimental results for both control typologies when no external load is applied.
From the experimental results, it can be seen that the overshoot is not present when LQR is employed. For the lowest reference value, the PID controller induces an overshoot of approximately 6% (
Figure 9a). Moreover, the previously observed dead-band is almost completely diminished in this case. Besides that, the faster switching time allows for a much faster overall system response for both PID and LQR controllers, which once more justifies the need of using the direct vacuum control principle if faster dynamics is desired. It can be noticed that the PMA has very similar dynamics for both control typologies, though PID again has slightly lower rising time constants for some reference values. Steady-state error is, however, always much lower in the case of LQR. LQR outperforms PID in terms of steady-state error by an order of magnitude in most cases (except for the 10 mm reference), while the error is a few tens of micrometers for all reference values.
Table 4 summarizes the values of rising times and steady-state errors.
On the other hand, when the results are compared to those of type A system (
Table 5), it can be observed that when the PID controller is used, the rising time constant is considerably lower for smaller reference values and slightly higher (4%) for the highest reference value when type B system is considered. The steady-state error is significantly lower for almost all references, with the exception of the lowest reference value (5 mm) where it is much higher in the case of the system with type B vacuum control. This behavior can be attributed to the highly nonlinear behavior of the analyzed pneumatic muscle.
Rising time constants are lower in all cases when using the LQR controller, and this is especially noticeable for lower reference values. Except for the highest reference value, steady-state error is again much lower in almost all cases.
Finally, the experimental results obtained by employing an experimental system with direct vacuum control (type B) allowed establishing significantly better results from a dynamical point of view. This is especially evident taking into account the fact that the dead-band effect during PMA activations is almost eliminated. Moreover, if compared to type A, the dynamical response is much faster especially for lower reference values. The steady-state error is in the case of LQR again several tens of micrometers (20–80 µm).
In order to test the muscle in more realistic conditions, we assess the positioning performances of the loaded system in the final set of experiments. The system is given a constant weight of 500 g. The results of the positioning performances for the loaded system are shown in
Figure 10, while the achieved dynamical performances are again evaluated in terms of rising time constants for each reference value, as shown in
Table 6. The graphs show that positioning performances without overshoot in the case of LQR and with slight overshoot for some references in the case of PID, are achieved. In both cases, a very small dead-band value is obtained at the beginning of the actuation cycle. When the rising time constants are compared to those achieved in the previous experiments, it can be concluded that the values are very similar and only differ by about 10%.
However, when using a PID controller, the steady-state error is much higher when the system is loaded, and this is especially evident for the lower references. This means that if the loading conditions change, the PID parameters have to be tuned again [
13]. This once more justifies the need for using more refined control typologies. When LQR with an additional pregain term is used, on the other hand, the steady-state error is a few tens of micrometers and it is not substantially influenced by external loading.
Finally, we conducted energy consumption analyses for the pneumatic systems under consideration by measuring the time required for the pressure in the compressor’s air reservoir to drop by 2 bar during the PMA operation. In both systems, the input pressure is held constant at 4 bar, and the control signal to the valves is sinusoidal with 0.01 Hz frequency. In these conditions, the total time measured was 870 and 264 s for type A (indirect vacuum control) and type B (direct vacuum control) systems, respectively. This allowed us to establish that the direct vacuum control system consumes 70% more compressed air than the indirect vacuum control approach. The direct vacuum control approach, however, allows better dynamical behavior, i.e., a faster response.